What this calculator does
The normal distribution describes an enormous range of natural and measured quantities: heights, measurement errors, test scores, manufacturing variation. Its ubiquity comes from the central limit theorem: sums of many small independent effects tend toward normality regardless of what the individual effects look like.
This calculator gives the area under the curve between two points, which is the probability of a value falling in that range. It also reports each tail separately and the familiar 68-95-99.7 boundaries.
The formula
Both bounds are converted to z-scores, then the cumulative distribution function is evaluated at each and subtracted. The result is the area under the curve between them, which for a probability distribution is exactly the probability of landing in that interval.
| Term | Meaning |
|---|---|
| μ | The mean: the centre of the distribution. |
| σ | The standard deviation: how wide it spreads. |
| Φ(z) | The cumulative distribution function: the area to the left of z. |
| 68-95-99.7 | The proportions within one, two and three standard deviations. |
The inputs explained
| Field | What to enter |
|---|---|
| Mean (μ) | The mean of the distribution. |
| Standard deviation (σ) | The standard deviation. Must be positive. |
| Lower bound | The lower bound of the interval. |
| Upper bound | The upper bound. |
When to use it
Estimating what proportion falls in a range
What share of adult heights fall between 165 and 180 cm, or what proportion of manufactured parts land within tolerance. Both are the same question.
Setting tolerance limits
Work backwards: adjust the bounds until the probability matches the yield you need. That gives the tolerance the process must hold.
Interpreting standardised tests
Test scores are usually designed to be approximately normal, so the proportion scoring in any band follows directly.
Checking the empirical rule
Set the bounds one standard deviation either side of the mean and the answer is about 68 per cent. Two gives about 95, three about 99.7: the rule most people half-remember, verified.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
Probability within a widening band
The interval widening symmetrically around a mean of 100 with a standard deviation of 15.
| Upper bound | P(a < X < b) | P(X < a) | P(X > b) |
|---|---|---|---|
| 100 | 34.1% | 15.9% | 50.0% |
| 110 | 58.9% | 15.9% | 25.2% |
| 115 | 68.3% | 15.9% | 15.9% |
| 130 | 81.9% | 15.9% | 2.28% |
| 145 | 84.0% | 15.9% | 0.135% |
| 160 | 84.1% | 15.9% | 0.003% |
How the spread changes the answer
A fixed interval from 90 to 120 with the standard deviation varying.
| Standard deviation | P(a < X < b) | P(X < a) | P(X > b) |
|---|---|---|---|
| 5 | 97.7% | 2.28% | 0.003% |
| 10 | 81.9% | 15.9% | 2.28% |
| 15 | 65.6% | 25.2% | 9.12% |
| 20 | 53.3% | 30.9% | 15.9% |
| 30 | 37.8% | 36.9% | 25.2% |
| 50 | 23.5% | 42.1% | 34.5% |
Questions
What is the 68-95-99.7 rule?
For a normal distribution, about 68 per cent of values fall within one standard deviation of the mean, 95 per cent within two and 99.7 per cent within three. It is a useful mental shortcut, and the calculator shows the exact boundaries.
How do I know if my data is normally distributed?
A histogram should look roughly symmetric and bell-shaped, and a Q-Q plot should be close to a straight line. Formal tests such as Shapiro-Wilk exist. Heavily skewed data, counts and bounded quantities are often not normal.
Can I calculate a one-tailed probability?
Yes: the P(X < a) and P(X > b) lines give exactly that. Set the relevant bound far enough away that it does not constrain the result.
Why does the normal distribution appear so often?
The central limit theorem: the sum of many independent random effects tends toward a normal distribution regardless of the individual distributions. Since most measured quantities are influenced by many small factors, normality emerges naturally.
What if my data is skewed?
The probabilities will be wrong, sometimes badly. Consider transforming the data, or use a distribution that fits better: lognormal for quantities bounded below at zero, for instance.
For a single value’s position, see the z-score calculator. For discrete counts, use binomial probability.